Quantum computing [1]} has the remarkable potential to dramatically surpass its classical counterpart on solving certain complex tasks in terms of the
processing speed or resource overhead. Universal quantum gates are crucial building blocks in quantum circuit model [2]}, [3]}, [4]}, [5]}, [6]}, quantum algorithms [7]}, [8]}, [9]}, quantum simulations [10]}, and quantum communication [11]}. Photon is generally viewed as one of the promising candidates for flying and solid-state quantum computing owing to its outstanding low decoherence, high-speed transmission, natural information carrier, flexible single-qubit manipulations, and available atom-like qubit interconnector [12]}, [13]}. Strong interactions between individual photons are the key resources for nontrivial multi-photon quantum gates, and the prohibited photon-photon interactions can be remedied efficiently by using linear optics [14]} or solid-state media [15]}, [16]}, [17]}. Unfortunately, solid-state platforms are challenged by inefficiency and imperfection in experiments. The probabilistic character of universal quantum gates with linear optics is unavoidable. Hence, minimizing the quantum resources required to implement quantum gates with higher success probability is a central problem of linear optical quantum computing, and tremendous efforts have been made on it [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}.
